Y is a subspace of a X which is a topology

Let Y be a subspace of X. If A is closed in Y and Y is closed in X, then A is closed in X.

First are there 2 cases to consider? (1) A is a proper subset of Y (2) or does this not matter?

Anyways, all I have is this:

Since A is closed in Y, is open in Y. Likewise, is open in X.

I am not sure what to do next.

Re: Y is a subspace of a X which is a topology

Quote:

Originally Posted by

**dwsmith** Let Y be a subspace of X. If A is closed in Y and Y is closed in X, then A is closed in X.

First are there 2 cases to consider? (1) A is a proper subset of Y (2)

or does this not matter?

Anyways, all I have is this:

Since A is closed in Y,

is open in Y. Likewise,

is open in X.

I am not sure what to do next.

By definition that is closed in you have that for some closed subset of ....so.

Re: Y is a subspace of a X which is a topology

Quote:

Originally Posted by

**Drexel28** By definition that

is closed in

you have that

for some closed subset

of

....so.

I don't know. All I can think of is to do then but I don't see how that will help.

Re: Y is a subspace of a X which is a topology

Ok, I think I have it.

where U is some closed set in X. Now, A is the intersection of two closed sets in X so A is closed in X.

Re: Y is a subspace of a X which is a topology

Quote:

Originally Posted by

**dwsmith** Ok, I think I have it.

where U is some closed set in X. Now, A is the intersection of two closed sets in X so A is closed in X.

Correct.